Duffing revisited: Phase-shift control and internal resonance in self-sustained oscillators

TL;DR

This paper investigates phase-shift control and internal resonance in the Duffing oscillator, revealing stable oscillations across the resonance curve and exploring frequency stabilization in micromechanical systems.

Contribution

It introduces phase-shift control as a means to stabilize oscillations over the entire resonance curve and analyzes internal resonance for frequency stabilization in micromechanical oscillators.

Findings

Oscillations are stable over the whole resonance curve under phase-shift control.

Internal resonance can stabilize oscillation frequency in micromechanical devices.

Preliminary experiments support the theoretical analysis.

Abstract

We address two aspects of the dynamics of the forced Duffing oscillator which are relevant to the technology of micromechanical devices and, at the same time, have intrinsic significance to the field of nonlinear oscillating systems. First, we study the stability of periodic motion when the phase shift between the external force and the oscillation is controlled -contrary to the standard case, where the control parameter is the frequency of the force. Phase-shift control is the operational configuration under which self-sustained oscillators -and, in particular, micromechanical oscillators- provide a frequency reference useful for time keeping. We show that, contrary to the standard forced Duffing oscillator, under phase-shift control oscillations are stable over the whole resonance curve. Second, we analyze a model for the internal resonance between the main Duffing oscillation mode and a higher-harmonic mode of a vibrating solid bar clamped at its two ends. We focus on the stabilization of the oscillation frequency when the resonance takes place, and present preliminary experimental results that illustrate the phenomenon. This synchronization process has been proposed to counteract the undesirable frequency-amplitude interdependence in nonlinear time-keeping micromechanical devices.